Analytical Solution for Vibrations of a Modified Timoshenko Beam on Visco-Pasternak Foundation Under Arbitrary Excitations

An analytical solution is derived for dynamic response of a modified Timoshenko beam with an infinite length resting on visco-Pasternak foundation subjected to arbitrary excitations. The modified Timoshenko beam model is employed to further consider the rotary inertia caused by the shear deformation of a beam, which is usually neglected by the traditional Timoshenko beam model. By using Fourier and Laplace transforms, the governing equations of motion are transformed from partial differential forms into algebraic forms in the Laplace domain. The analytical solution is then converted into the time domain by applying inverse transforms and convolution theorem. Some widely used loading cases, including moving line loads for nondestructive testing, travelling loads for seismic wave passage, and impulsive load for impact vibration, are also discussed in this paper. The proposed generic solutions are verified by comparing their degraded results to the known solutions in other literature. Several examples are performed to further investigate the differences of the beam responses obtained from the modified and the traditional Timoshenko beam models. Results show that the modified Timoshenko beam simulates the beam responses more accurately than the traditional model, especially under the dynamic loads with a high frequency. The analytical solutions proposed in this paper can be conveniently used for design and applied as an effective tool for practitioners.

Geometrical Nonlinearity for a Timoshenko Beam with Flexoelectricity

The Timoshenko beam model is applied to the analysis of the flexoelectric effect for a cantilever beam under large deformations. The geometric nonlinearity with von Kármán strains is considered. The nonlinear system of ordinary differential equations (ODE) for beam deflection and rotation are derived. Moreover, this nonlinear system is linearized for each load increment, where it is solved iteratively. For the vanishing flexoelectric coefficient, the governing equations lead to the classical Timoshenko beam model. Furthermore, the influence of the flexoelectricity coefficient and the microstructural length-scale parameter on the beam deflection and the induced electric intensity is investigated.

AN EFFICIENT FINITE DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION OF TIMOSHENKO BEAM MODEL

We propose and implement a finite difference scheme for the numerical solution of the Timoshenko beam model without locking phenomenon. The averaging concept is used in approximating the function, and thus developing the scheme for elements. Finally, the system is discretized into the algebraic system using the proposed scheme and the numerical solution is attained. The numerical solutions are attained for a constant load and a variable load comprising linear and exponential functions. The mathematical model of the Timoshenko beam (TB) problem in the form of a boundary-value problem has been solved successfully for the rotation and displacement parameters. The results agree with other schemes in the literature for various values of the parameter and step size.

ON EXACT ANALYTICAL SOLUTIONS OF THE TIMOSHENKO BEAM MODEL UNDER UNIFORM AND VARIABLE LOADS

In this research work, we consider the mathematical model of the Timoshenko beam (TB) problem in the form of a boundary-value problem of a system of ordinary differential equations. Instead of numerical solution using finite difference and finite volume methods, an attempt is made to derive the exact analytical solutions of the model with boundary feedback for a better and explicit description of the rotation and displacement parameters of the TB structure model. The explicit analytical solutions have been successfully found for the uniform and real-time variable load cases. The rotation and displacement profiles obtained through the analytical solutions accurately picture the structure of the beam under uniform and variable loads.

Free vibration of a single-walled carbon nanotube based on the nonlocal Timoshenko beam model

Free vibration of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium is studied on the basis of the nonlocal Timoshenko beam model. Influences of the slenderness ratios, the boundary conditions, the atomic structures and the stiffness of the embedded medium on the natural frequencies and mode shapes of SWCNT are examined. The nonlocal effect is significant for the higher modes of SWCNT with a small slenderness ratio embedded in a soft elastic medium, and it softens the SWCNT except for the fundamental frequency of the clamped–free SWCNT.